Strange characters after site migration

A client of mine decided to change from one hosting company to another. The previous host provided an SQL backup and all of the site files.

Once all the site files were uploaded to the new server and the DB was restored, we began having the following characters appear on the site:
?�?��?��

I visited the forums to try to find an answer to this issue, and found information regarding page encoding. The site is currently set to “UTF-8”, but changing this to “ISO-8859-1”, “UTF-16” and even “Windows-1252” seems to have little effect. Changing to “ISO-8859-1” actually changed the strange black diamonds with question marks to:
?ó?é¼?äól

I then thought to check the source and see if perhaps the DB had strange characters contained within the posts. What I found was that the DB contained the following code for apostrophes:
ॳóΓé¼Γäó

These strange characters seem to be appearing for all apostrophes, right and left quotation marks, mdashes, etc.

Any clue what I can do to resolve the issue that doesn’t result in going into every single post and comment and changing the character manually? I’ve already tried the UTF8 Sanitize plugin without any luck.